A263255 -id:A263255 - cp's OEIS frontend

A263274 One-based column index to array A263255.

1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 2, 5, 1, 6, 2, 6, 1, 3, 4, 5, 1, 6, 1, 7, 1, 5, 2, 8, 1, 2, 1, 7, 2, 7, 1, 9, 1, 6, 2, 10, 2, 7, 8, 8, 1, 9, 1, 11, 2, 9, 1, 12, 1, 10, 1, 13, 2, 3, 1, 10, 1, 11, 2, 14, 1, 4, 1, 12, 2, 13, 1, 15, 1, 14, 8, 15, 11, 3, 12, 9, 5, 10, 4, 16, 5, 13, 2

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10002

Cf. A259934, A263254, A263255.

Cf. also A263275.

(define (A263274 n) (let ((rowf (rowfun-for-A263255 (A263254 n)))) (let loop ((i 1)) (if (= n (rowf i)) i (loop (+ 1 i))))))
;; rowfun-for-A263255 given in A263255.

Other identities. For all n >= 0:

a(A259934(n)) = n+1.

A263275 One-based row-index to array A263255.

Original entry on oeis.org

1, 2, 1, 3, 3, 4, 1, 5, 4, 2, 2, 3, 1, 4, 3, 4, 4, 5, 1, 6, 4, 6, 1, 7, 5, 2, 2, 8, 2, 9, 1, 10, 3, 10, 1, 11, 9, 12, 2, 12, 4, 13, 1, 14, 3, 13, 1, 14, 3, 2, 4, 15, 2, 16, 1, 16, 4, 17, 1, 18, 4, 19, 1, 18, 18, 20, 2, 21, 2, 21, 1, 22, 5, 23, 2, 22, 2, 24, 1, 25, 2, 3, 2, 4, 6, 4, 3, 5, 3, 6, 1, 6, 4, 7, 1, 7

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10002

One more than A263254.
Cf. A049820, A262693, A263255.
Cf. also A263274.

If A262693(n) = 1, then a(n) = 1, otherwise a(n) = 1 + a(A049820(n)).

a(n) = 1 + A263254(n).

A263256 Transpose of array A263255.

Original entry on oeis.org

0, 1, 2, 3, 9, 6, 5, 4, 10, 12, 7, 8, 11, 25, 18, 19, 17, 13, 14, 26, 22, 23, 21, 24, 15, 32, 28, 30, 27, 93, 84, 72, 16, 44, 38, 34, 29, 97, 95, 89, 87, 20, 48, 49, 42, 31, 36, 99, 96, 91, 98, 40, 81, 52, 46, 35, 33, 101, 104, 100, 143, 139, 50, 86, 66, 54, 37, 109, 103, 105, 108, 149, 145, 141, 56, 88, 68, 58

Offset: 0

Antti Karttunen, Nov 07 2015

See comments in A263255.

			Top left corner of the array:
    0,   1,   3,   5,   7,  19,  23,  27,  29,  31,  35,  37,  41,  43,  51
    2,   9,   4,   8,  17,  21,  93,  97,  36,  33, 109,  39,  45,  47, 179
    6,  10,  11,  13,  24,  84,  95,  99, 101, 103, 111, 113, 119, 123, 413
   12,  25,  14,  15,  72,  89,  96, 104, 105, 107, 136, 115, 173, 177, 415
   18,  26,  32,  16,  87,  91, 100, 108, 120, 128, 140, 117, 407, 409, 479
   22,  28,  44,  20,  98, 143, 149, 151, 155, 132, 161, 163, 471, 411, 481
   30,  38,  48,  40, 139, 145, 153, 158, 164, 157, 165, 167, 473, 477, 495
   34,  49,  81,  50, 141, 146, 154, 224, 232, 159, 393, 171, 475, 483, 581
   42,  52,  86,  56, 142, 147, 160, 248, 260, 276, 395, 397, 568, 484, 583
   46,  66,  88,  60, 150, 148, 212, 307, 264, 280, 399, 401, 571, 577, 587
   54,  68, 116,  83, 202, 206, 229, 309, 311, 313, 461, 403, 572, 578, 589
   58,  74, 130,  85, 208, 220, 240, 379, 324, 389, 465, 405, 573, 579, 594
   62,  76, 135,  92, 225, 227, 305, 381, 383, 391, 532, 463, 575, 580, 671
   70,  80, 175, 134, 228, 301, 356, 385, 387, 453, 540, 467, 659, 585, 676
   78,  82, 184, 137, 297, 350, 364, 447, 400, 457, 567, 469, 663, 661, 677
   90, 110, 186, 138, 338, 352, 377, 458, 449, 459, 584, 544, 665, 667, 679
   94, 112, 192, 198, 340, 373, 443, 464, 451, 500, 600, 560, 861, 669, 869
  102, 122, 200, 204, 371, 440, 445, 480, 455, 516, 651, 569, 874, 675, 871
  106, 124, 244, 216, 372, 441, 454, 536, 472, 520, 851, 616, 904, 863, 873
  ...

Cf. A263255.

(define (A263256 n) (A263255bi (A025581 n) (+ 1 (A002262 n))))
;; Other code as in A263255.

A263262 Row 2 of A263255.

Original entry on oeis.org

3, 4, 11, 14, 32, 44, 48, 81, 86, 88, 116, 130, 135, 175, 184, 186, 192, 200, 244, 246, 252, 262, 266, 284, 286, 296, 320, 326, 336, 365, 370, 376, 378, 380, 384, 404, 406, 428, 430, 460, 482, 490, 524, 529, 534, 608, 624, 642, 652, 654, 656, 682, 690, 764, 768, 772, 786, 794, 796, 802, 806, 810, 884, 888

Offset: 1

Antti Karttunen, Nov 07 2015

nonn

Numbers n for which A263254(n) = 2. Numbers that are two edges removed from the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Antti Karttunen, Table of n, a(n) for n = 1..9096

Cf. A049820, A259934, A263254, A263255.

A263261 Row 1 of A263255.

Original entry on oeis.org

1, 9, 10, 25, 26, 28, 38, 49, 52, 66, 68, 74, 76, 80, 82, 110, 112, 122, 124, 126, 127, 131, 133, 156, 168, 169, 170, 172, 176, 178, 180, 188, 218, 226, 234, 238, 258, 278, 288, 290, 300, 304, 306, 316, 318, 322, 334, 361, 362, 368, 398, 414, 422, 448, 452, 470, 478, 496, 514, 518, 526, 586

Offset: 1

Antti Karttunen, Nov 07 2015

nonn

Numbers n for which A263254(n) = 1. Numbers that are one edge removed from the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A049820, A259934, A263254, A263255.

Cf. also A262901.

A263263 Row 3 of A263255.

Original entry on oeis.org

5, 8, 13, 15, 16, 20, 40, 50, 56, 60, 83, 85, 92, 134, 137, 138, 198, 204, 216, 268, 272, 289, 292, 308, 312, 328, 332, 344, 348, 360, 367, 382, 392, 396, 412, 416, 436, 438, 498, 533, 538, 620, 632, 636, 640, 664, 666, 694, 700, 704, 720, 776, 792, 814, 816, 828, 922, 936, 940, 956, 965, 966, 969, 1032

Offset: 1

Antti Karttunen, Nov 07 2015

nonn

Numbers n for which A263254(n) = 3. Numbers that are three edges removed from the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Antti Karttunen, Table of n, a(n) for n = 1..2047

Cf. A049820, A259934, A263254, A263255.

A263264 Row 4 of A263255.

Original entry on oeis.org

7, 17, 24, 72, 87, 98, 139, 141, 142, 150, 202, 208, 225, 228, 297, 338, 340, 371, 372, 375, 386, 388, 408, 420, 424, 432, 502, 506, 537, 539, 542, 550, 625, 644, 660, 680, 684, 698, 712, 732, 804, 818, 822, 824, 926, 948, 952, 967, 968, 971, 973, 976, 990, 1056, 1062, 1120, 1122

Offset: 1

Antti Karttunen, Nov 07 2015

nonn

Numbers n for which A263254(n) = 4. Numbers that are four edges removed from the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Antti Karttunen, Table of n, a(n) for n = 1..1023

Cf. A049820, A259934, A263254, A263255.

A049820 a(n) = n - d(n), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 5, 4, 6, 6, 9, 6, 11, 10, 11, 11, 15, 12, 17, 14, 17, 18, 21, 16, 22, 22, 23, 22, 27, 22, 29, 26, 29, 30, 31, 27, 35, 34, 35, 32, 39, 34, 41, 38, 39, 42, 45, 38, 46, 44, 47, 46, 51, 46, 51, 48, 53, 54, 57, 48, 59, 58, 57, 57, 61, 58, 65

Offset: 1

Clark Kimberling

a(n) is the number of non-divisors of n in 1..n. - Jaroslav Krizek, Nov 14 2009
Also equal to the number of partitions p of n such that max(p)-min(p) = 1. The number of partitions of n with max(p)-min(p) <= 1 is n; there is one with k parts for each 1 <= k <= n. max(p)-min(p) = 0 iff k divides n, leaving n-d(n) with a difference of 1. It is easiest to see this by looking at fixed k with increasing n: for k=3, starting with n=3 the partitions are [1,1,1], [2,1,1], [2,2,1], [2,2,2], [3,2,2], etc. - Giovanni Resta, Feb 06 2006 and Franklin T. Adams-Watters, Jan 30 2011
Number of positive numbers in n-th row of array T given by A049816.
Number of proper non-divisors of n. - Omar E. Pol, May 25 2010
a(n+2) is the sum of the n-th antidiagonal of A225145. - Richard R. Forberg, May 02 2013
For n > 2, number of nonzero terms in n-th row of triangle A051778. - Reinhard Zumkeller, Dec 03 2014
Number of partitions of n of the form [j,j,...,j,i] (j > i). Example: a(7)=5 because we have [6,1], [5,2], [4,3], [3,3,1], and [2,2,2,1]. - Emeric Deutsch, Sep 22 2016

			a(7) = 5; the 5 non-divisors of 7 in 1..7 are 2, 3, 4, 5, and 6.
The 5 partitions of 7 with max(p) - min(p) = 1 are [4,3], [3,2,2], [2,2,2,1], [2,2,1,1,1] and [2,1,1,1,1,1]. - _Emeric Deutsch_, Mar 01 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. E. Andrews, M. Beck, N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv preprint arXiv:1406.3374 [math.NT], 2014-2015.

Cf. A000005.
One less than A062968, two less than A059292.
Cf. A161664 (partial sums).
Cf. A060990 (number of solutions to a(x) = n).
Cf. A045765 (numbers not occurring in this sequence).
Cf. A236561 (same sequence sorted into ascending order), A236562 (with also duplicates removed), A236565, A262901 and A262903.
Cf. A262511 (numbers that occur only once).
Cf. A055927 (positions of repeated terms).
Cf. A245388 (positions of squares).
Cf. A155043 (number of steps needed to reach zero when iterating a(n)), A262680 (number of nonzero squares encountered).
Cf. A259934 (an infinite trunk of the tree defined by edge-relation a(child) = parent, conjectured to be unique).
Cf. tables and arrays A047916, A051731, A051778, A173540, A173541.
Cf. also arrays A225145, A262898, A263255 and tables A263265, A263267.
Other related sequences: A006218, A006590, A051953, A070824, A094181, A062249, A067391, A076627, A128508, A131187, A134156, A140826, A161886, A177235, A177236, A227874, A228453, A230653, A230654, A231167, A245197, A253473.

List([1..80],n->n-Tau(n)); # Muniru A Asiru, Sep 28 2018

a049820 n = n - a000005 n  -- Reinhard Zumkeller, Feb 06 2012

A049820 := n->n-numtheory[tau](n):
seq(A049820(n), n=1..100);

Table[n - DivisorSigma[0, n], {n, 100}] (* Wesley Ivan Hurt, Nov 19 2014 *)
Array[(# - DivisorSigma[0, #])&, 70] (* Vincenzo Librandi, Dec 29 2015 *)

PARI
```
a(n)=n-numdiv(n)
```

(define (A049820 n) (- n (A000005 n))) ;; Antti Karttunen, Nov 27 2015

a(n) = Sum_{k=1..n} ceiling(n/k)-floor(n/k). - Benoit Cloitre, May 11 2003
G.f.: Sum_{k>0} x^(2*k+1)/(1-x^k)/(1-x^(k+1)). - Emeric Deutsch, Mar 01 2006
a(n) = A006590(n) - A006218(n) = A161886(n) - A000005(n) - A006218(n) + 1 for n >= 1. - Jaroslav Krizek, Nov 14 2009
a(n) = Sum_{k=1..n} A000007(A051731(n,k)). - Reinhard Zumkeller, Mar 09 2010
a(n) = A076627(n) / A000005(n). - Reinhard Zumkeller, Feb 06 2012
For n >= 2, a(n) = A094181(n) / A051953(n). - Antti Karttunen, Nov 27 2015
a(n) = Sum_{k=1..n} ((n mod k) + (-n mod k))/k. - Wesley Ivan Hurt, Dec 28 2015
G.f.: Sum_{j>=2} (x^(j+1)*(1-x^(j-1))/(1-x^j))/(1-x). - Emeric Deutsch, Sep 22 2016
Dirichlet g.f.: zeta(s-1)- zeta(s)^2. - Ilya Gutkovskiy, Apr 12 2017
a(n) = Sum_{i=1..n-1} sign(i mod n-i). - Wesley Ivan Hurt, Sep 27 2018

Edited by Franklin T. Adams-Watters, Jan 30 2012

A259934 Infinite sequence starting with a(0)=0 such that A049820(a(k)) = a(k-1) for all k>=1, where A049820(n) = n - (number of divisors of n).

Original entry on oeis.org

0, 2, 6, 12, 18, 22, 30, 34, 42, 46, 54, 58, 62, 70, 78, 90, 94, 102, 106, 114, 118, 121, 125, 129, 144, 152, 162, 166, 174, 182, 190, 194, 210, 214, 222, 230, 236, 242, 250, 254, 270, 274, 282, 294, 298, 302, 310, 314, 330, 342, 346, 354, 358, 366, 374, 390, 394, 402, 410, 418, 426, 434, 442, 446, 462, 466, 474, 486, 494, 510, 522, 530, 546, 558, 562, 566, 574, 582, 590

Offset: 0

Max Alekseyev, Jul 09 2015

Equivalently, satisfies the property: A000005(a(n)) = a(n)-a(n-1). The first differences a(n)-a(n-1) are given in A259935.
V. S. Guba (2015) proved that such an infinite sequence exists. Numerical evidence suggests that it may also be unique -- is it? All terms below 10^10 are defined uniquely.
If the current definition does not uniquely define the sequence, the "lexicographically earliest" condition may be added to make the sequence well-defined.
From Vladimir Shevelev, Jul 21 2015: (Start)
If a(k), a(k+1), a(k+2) is an arithmetic progression, then a(k+1) is in A175304.
Indeed, by the definition of this sequence, a(n)-a(n-1) = d(a(n)), for all n>=1, where d(n) = A000005(n). Hence, have a(k+1) - a(k) = a(k+2) - a(k+1) = d(a(k+1)) = d(a(k+2)). So a(k+1) + d(a(k+2)) = a(k+2) and a(k+1) + d(a(k+1)) = a(k+2).
Therefore, d(a(k+1) + d(a(k+1))) = d(a(k+2))= d(a(k+1)), i.e., a(k+1) is in A175304. Thus, if there are infinitely many pairs of the same consecutive terms of A259935, then A175304 is infinite (see there my conjecture). (End)
From Antti Karttunen, Nov 27 2015: (Start)
If multiple apparently infinite branches would occur at some point of computing, then even if the "lexicographically earliest" condition were then added to the definition, it would not help us much (when computing the sequence), as we would still not know which of the said branches were truly infinite. [See also Max Alekseyev's latter Jul 9 2015 posting on SeqFan-list, where he notes the same thing.] Note that many of the derived sequences tacitly assume that the uniqueness-conjecture is true. See also comments at A262693 and A262896.
One sufficient (but not a necessary) condition for the uniqueness of this sequence is that the sequence A262509 has infinite number of terms. Please see further comments there.
The graph of sequence exhibits two markedly different slopes, depending on whether it is on the "fast lane" of A049820 (even numbers) or the "slow lane" [odd numbers, for example when traversing the 1356 odd terms from 123871 to 113569 at range a(9859) .. a(8504)]. See A263086/A263085 for the "average cumulative speed difference" between the lanes. In general, slow and fast lane stay separate, except when they terminate into one of the squares (A262514) that work as "exchange ramps", forcing the parity (and thus the speed) to change. In average, the odd squares are slightly better than the even squares in attracting lanes going towards smaller numbers (compare A263253 to A263252). The cumulative effect of this bias is that the odd terms are much rarer in this sequence than the even terms (compare A263278 to A262516).
(End)

Robert Israel, Table of n, a(n) for n = 0..64800
M. Alekseyev et al., Apparently unique infinite sequences related to the sum of divisors, Discussion in SeqFan mailing list, 2015.
Michael De Vlieger, Poster illustrating A259934 and A263267
V. S. Guba et al., Sequence and divisors, 2015. (in Russian)

Cf. A000005, A049820, A060990, A259935 (first differences).
Topmost row of A263255. Cf. also irregular tables A263267 & A263265 and array A262898.
Cf. A262693 (characteristic function).
Cf. A155043, A262694, A262904 (left inverses).
Cf. A262514 (squares present), A263276 (their positions), A263277.
Cf. A262517 (odd terms).
Cf. A262509, A262510, A262897 (other subsequences).
Cf. also A175304, A260257, A262680.
Cf. A261089, A261103, A262503, A262506, A262516, A263279, A263280, A263085, A263086, A263253, A263257, A263278.
Cf. also A262679, A262896 (see the C++ program there).
No common terms with A045765 or A262903.
Positions of zeros in A262522, A262695, A262696, A262697, A263254.
Various metrics concerning finite side-trees: A262888, A262889, A262890.
Cf. also A262891, A262892 and A262895 (cf. its graph).
Cf. A260084, A260124 (variants).
Cf. also A179016 (a similar "beanstalk trunk sequence" but with more tractable and regular behavior).

N:= 10^4: # to get "guaranteed unique" terms <= N
S:= Vector(N,datatype=integer[1]):
for n from N+1 to 2*N do
  k:= n - numtheory:-tau(n);
  if k <= N then S[k]:= S[k]+1; B[k]:= n; fi;
od:
for n from N to 3 by -1 do
  if S[n] >= 1 then
    k:= n - numtheory:-tau(n);
    S[k]:= S[k]+1; B[k]:= n;
  fi
od:
A[0]:= 0: A[1]:= 2:
for n from 2 do
  b:= B[A[n-1]];
  if b > N or S[b] > 1 then break fi;
  A[n]:= b;
od:
seq(A[i],i=0..n-1); # Robert Israel, Jul 09 2015

NN = 10^4; (* to get "guaranteed unique" terms <= NN *)
Clear[A, B, S]; S[]=0; For[n = NN+1, n <= 2*NN, n++, k = n-DivisorSigma[0, n]; If[k <= NN, S[k] = S[k]+1; B[k]=n]]; For[n=NN, n >= 3, n--, If[S[n] >= 1 , k = n-DivisorSigma[0, n]; S[k] = S[k]+1; B[k]=n]]; A[0]=0; A[1]=2; For[n=2, True, n++, b = B[A[n-1]]; If[b>NN || S[b]>1, Break[]]; A[n]=b]; Table[A[i], {i, 0, n-1}] (* _Jean-François Alcover, Jul 22 2015, after Robert Israel *)

From Antti Karttunen, Nov 27 2015: (Start)
Other identities and observations. For all n >= 0:
a(n) = A262679(A262896(n)).
A155043(a(n)) = A262694(a(n)) = A262904(a(n)) = n.
A261089(n) <= a(n) <= A262503(n). [A261103 and A262506 give the distances of a(n) to these bounds.]
(End)

A263265 Irregular triangle T(n,k), n >= 0, k = 1 .. A262507(n), read by rows, where each row n lists in ascending order all integers x for which A155043(x) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 8, 9, 10, 12, 7, 11, 14, 18, 13, 15, 16, 20, 22, 17, 24, 25, 26, 28, 30, 19, 21, 32, 34, 23, 38, 40, 42, 27, 44, 46, 48, 29, 36, 49, 50, 52, 54, 56, 60, 31, 33, 58, 72, 35, 62, 66, 84, 37, 39, 68, 70, 96, 41, 45, 74, 76, 78, 80, 104, 108, 43, 47, 81, 82, 88, 90, 120, 51, 83, 85, 86, 94, 128, 132, 53, 55, 87, 92, 102, 136, 140

Offset: 0

Antti Karttunen, Nov 24 2015

			Rows 0 - 8 of the triangle:
0;
1, 2;
3, 4, 6;
5, 8, 9, 10, 12;
7, 11, 14, 18;
13, 15, 16, 20, 22;
17, 24, 25, 26, 28, 30;
19, 21, 32, 34;
23, 38, 40, 42;
Row n contains A262507(n) terms, the first of which is A261089(n) and the last of which is A262503(n). For all terms on row n, A155043(n) = n.

Inverse: A263266.
Cf. A261089 (left edge), A262503 (right edge), A262507 (number of terms on each row).
Cf. A263279 (gives the positions of terms of A259934 on each row), A263280 (and their distance from the right edge).
Cf. A155043, A263259, A263270.
Cf. also permutations A263267 & A263268 and A263255 & A263256.
Differs from A263267 for the first time at n=31, where a(31) = 38, while A263267(31) = 40.

Other identities. For all n >= 0:

A155043(a(n)) = A263270(n).

cp's OEIS Frontend

A263274 One-based column index to array A263255.

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A263275 One-based row-index to array A263255.

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A263256 Transpose of array A263255.

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A263262 Row 2 of A263255.

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A263261 Row 1 of A263255.

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A263263 Row 3 of A263255.

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A263264 Row 4 of A263255.

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A049820 a(n) = n - d(n), where d(n) is the number of divisors of n (A000005).

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A259934 Infinite sequence starting with a(0)=0 such that A049820(a(k)) = a(k-1) for all k>=1, where A049820(n) = n - (number of divisors of n).

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